The fact that the Abel map does not effect a linearization of the divisor
\(\mathcal {D}_{\hat{\mu }_0(x,t_r)\underline{\hat{\mu }}(x,t_r)}\)  in the
time-dependent HS2 context, which is well known and discussed
(using different approaches)
by Constantin and McKean [1]}, Alber, Camassa, Fedorov, Holm,
and Marsden [2]}, Alber and Fedorov [3]}, [4]}.
The change of variables
\( x \mapsto \tilde{x}= \int ^x dx^\prime \left(\frac{4a}{\Psi _{n+1}(\bar{\mu }(x^\prime ,t_r))}\right)\) 
